The influence function has also been used recently in Machine Learning literature in order to have a model selection tool specialized in robustness, see for example [1]}, [2]} and the closely related tool of leave one out error [3]}.
The field of Robustness in Machine Learning has been very active in the last few years, in particular after several works by Olivier Catoni and co-authors in [4]}, [5]}, the goal being to prove non-asymptotic deviation bounds when the data are more heavy-tailed than what is usually considered in classical Machine Learning. This line of thought has been continued in a number of articles, in particular [6]} introduced some general concept of sub-Gaussian estimators that have been then used successfully in other applications, see [7]}, [8]}, [4]}, [10]}, [11]}, [12]}. See also some comprehensive lecture notes on the subject in [13]}.
